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Difference between revisions of "Division ring"

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m (Richard Pinch moved page Ring with division to Division ring: more usual title)
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''division ring''
 
  
A (not necessarily associative) [[Ring|ring]] in which the equations
 
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082440/r0824401.png" /></td> </tr></table>
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A ring with division is a (not necessarily associative) [[ring]] in which the equations
  
are solvable for any two elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082440/r0824402.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082440/r0824403.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082440/r0824404.png" />. If the solutions of these equations are uniquely determined, then the ring is called a quasi-division ring. In contrast to an arbitrary division ring, a quasi-division ring cannot have divisors of zero (cf. [[Zero divisor|Zero divisor]]); the non-zero elements of a quasi-division ring form a [[Quasi-group|quasi-group]] with respect to multiplication. Each (not necessarily associative) ring without divisors of zero can be imbedded in a quasi-division ring. An associative division ring is an (associative) [[Skew-field|skew-field]]. See also [[Division algebra|Division algebra]].
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$$a \cdot x = b \quad;\quad\quad y \cdot a = b $$
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are solvable for any two elements $a$ and $b$, where $a \ne 0$. If the solutions of these equations are uniquely determined, then the ring is called a quasi-division ring. In contrast to an arbitrary division ring, a quasi-division ring cannot have divisors of zero (cf. [[Zero divisor|Zero divisor]]); the non-zero elements of a quasi-division ring form a [[Quasi-group|quasi-group]] with respect to multiplication. Each (not necessarily associative) ring without divisors of zero can be imbedded in a quasi-division ring. An associative division ring is an (associative) [[Skew-field|skew-field]]. See also [[Division algebra|Division algebra]].
  
 
====References====
 
====References====

Revision as of 08:30, 28 December 2013


A ring with division is a (not necessarily associative) ring in which the equations

$$a \cdot x = b \quad;\quad\quad y \cdot a = b $$


are solvable for any two elements $a$ and $b$, where $a \ne 0$. If the solutions of these equations are uniquely determined, then the ring is called a quasi-division ring. In contrast to an arbitrary division ring, a quasi-division ring cannot have divisors of zero (cf. Zero divisor); the non-zero elements of a quasi-division ring form a quasi-group with respect to multiplication. Each (not necessarily associative) ring without divisors of zero can be imbedded in a quasi-division ring. An associative division ring is an (associative) skew-field. See also Division algebra.

References

[1] A.G. Kurosh, "Lectures on general algebra" , Chelsea (1963) (Translated from Russian)


Comments

References

[a1] N. Jacobson, "The theory of rings" , Amer. Math. Soc. (1943)
How to Cite This Entry:
Division ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Division_ring&oldid=31217
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article